Non-hermitian random matrix theory: Method of hermitian reduction

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We consider random non-hermitian matrices in the large-N limit. The power of analytic function theory cannot be brought to bear directly to analyze non-hermitian random matrices, in contrast to hermitian random matrices. To overcome this difficulty, we show that associated to each ensemble of non-hermitian matrices there is an auxiliary ensemble of random hermitian matrices which can be analyzed by the usual methods. We then extract the Green function and the density of eigenvalues of the non-hermitian ensemble from those of the auxiliary ensemble. We apply this "method of hermitization" to several examples, and discuss a number of related issues.

Original languageEnglish
Pages (from-to)579-608
Number of pages30
JournalNuclear Physics B
Issue number3
StatePublished - 3 Nov 1997
Externally publishedYes

Bibliographical note

Funding Information:
A.Z. would like to thank David Nelson for stimulating his interest in non-hermitian random matrices. He also thanks Edouard BrEzin for extensive discussions and the l~cole Normale Sup6rieure for its hospitality. Some of the calculations reported here were first done with the help of Edouard Br6zin. A.Z. also thanks Freeman Dyson for helpful conversations and the Institute for Advanced Study for a Dyson Distinguished Visiting Professorship. J.E would like to thank Gerald Dunne for some references concerning quaternionic analysis. Both authors are obliged to Mark Srednicki for help with the numerical calculations. This work was supported in part by the National Science Foundation under Grant No. PHY89-04035, and by the Dyson Visiting Professor Funds.


  • Deterministic plus random
  • Non-gaussian ensembles
  • Non-hermitian hamiltonians and localization
  • Non-hermitian random matrices
  • The method of hermitization

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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